2,161 research outputs found
The edge chromatic number of outer-1-planar graphs
A graph is outer-1-planar if it can be drawn in the plane so that all
vertices are on the outer face and each edge is crossed at most once. In this
paper, we completely determine the edge chromatic number of outer 1-planar
graphs
Implementing path coloring algorithms on planar graphs
Master's Project (M.S.) University of Alaska Fairbanks, 2017A path coloring of a graph partitions its vertex set into color classes such that each class induces a disjoint union of paths. In this project we implement several algorithms to compute path colorings of graphs embedded in the plane. We present two algorithms to path color plane graphs with 3 colors based on a proof by Poh in 1990. First we describe a naive algorithm that directly follows Poh's procedure, then we give a modified algorithm that runs in linear time. Independent results of Hartman and Skrekovski describe a procedure that takes a plane graph G and a list of 3 colors for each vertex, and computes a path coloring of G such that each vertex receives a color from its list. We present a linear time implementation based on Hartman and Skrekovski's proofs. A C++ implementation is provided for all three algorithms, utilizing the Boost Graph Library. Instructions are given on how to use the implementation to construct colorings for plane graphs represented by Boost data structures
5-list-coloring planar graphs with distant precolored vertices
We answer positively the question of Albertson asking whether every planar
graph can be -list-colored even if it contains precolored vertices, as long
as they are sufficiently far apart from each other. In order to prove this
claim, we also give bounds on the sizes of graphs critical with respect to
5-list coloring. In particular, if G is a planar graph, H is a connected
subgraph of G and L is an assignment of lists of colors to the vertices of G
such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G
contains a subgraph with O(|H|^2) vertices that is not L-colorable.Comment: 53 pages, 9 figures version 2: addresses suggestions by reviewer
Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count
We show that triangle-free penny graphs have degeneracy at most two, list
coloring number (choosability) at most three, diameter , and
at most edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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